What is the proof of the theorem: [x] + [x + 1/n] + [x + 2/n] + ... + [x + (n - 2)/n] + [x + (n - 2)/n][x + (n - 1)/n] = [nx]

Solution:

[.] represents the Greatest Integer Function.

The greatest integer function implies [a  + (p/q)] = a (a be any integer and (p/q) < 1)

L.H.S. = [x] + [x + 1/n] + [x + 2/n] + … + [x + (n - 2)/n] + [x + (n - 2)/n] + [x + (n - 1)/n] 

After simplifying it by the greatest integer property that,

[a  + (p/q)] = a (a be any integer and (p/q) < 1)

L.H.S. =  [x] + [x] + [x] + ...+ till n times

So, L.H.S. = n[x] = [nx] = R.H.S.

Therefore, [x] + [x + 1/n] + [x + 2/n] + ……. + [x + (n - 2)/n] + [x + (n - 2)/n][x + (n - 1)/n] = [nx]

What is the proof of the theorem: [x] + [x + 1/n] + [x + 2/n] + ... + [x + (n - 2)/n] + [x + (n - 2)/n][x + (n - 1)/n] = [nx]

Summary:

Hence, proved [x] + [x + 1/n] + [x + 2/n] + ... + [x + (n - 2)/n] + [x + (n - 2)/n][x + (n - 1)/n] = [nx]